Determine the general solution of $2 \sin 2 x+3 \sin x=0$
Answer (1)
Use the double angle identity to rewrite the equation: 2 ( 2 sin x cos x ) + 3 sin x = 0 .
Factor out sin x : sin x ( 4 cos x + 3 ) = 0 .
Solve for sin x = 0 , which gives x = nπ , where n is an integer.
Solve for cos x = − 4 3 , which gives x = 2 nπ ± arccos ( − 4 3 ) , where n is an integer. The final answer is x = nπ , 2 nπ ± arccos ( − 4 3 ) .
Explanation
Problem Analysis We are given the equation 2 sin 2 x + 3 sin x = 0 and we want to find the general solution.
Using Double Angle Identity We can use the double angle identity sin 2 x = 2 sin x cos x to rewrite the equation as: 2 ( 2 sin x cos x ) + 3 sin x = 0
Simplifying Simplifying the equation, we get: 4 sin x cos x + 3 sin x = 0
Factoring Now, we factor out sin x :
sin x ( 4 cos x + 3 ) = 0
Two Cases This gives us two possible cases:
Case 1: sin x = 0
Case 2: 4 cos x + 3 = 0
Solving Case 1 For Case 1, the general solution for sin x = 0 is: x = nπ , n ∈ Z
Solving Case 2 For Case 2, we have: 4 cos x + 3 = 0 cos x = − 4 3
Finding General Solution for Case 2 To find the general solution for cos x = − 4 3 , we first find the reference angle α such that cos − 1 ( 4 3 ) ≈ 0.7227 radians. Since cos x is negative, x lies in the second and third quadrants.
The general solution is given by: x = 2 nπ ± arccos ( − 4 3 ) , n ∈ Z Using a calculator, arccos ( − 4 3 ) ≈ 2.4189 radians.
Combining Solutions Therefore, the general solution for the given equation is: x = nπ or x = 2 nπ ± arccos ( − 4 3 ) , n ∈ Z
We can approximate arccos ( − 4 3 ) as 2.4189 radians.
Examples
Trigonometric equations like this one are used in physics to model oscillations and wave phenomena. For example, the motion of a pendulum or the behavior of alternating current in an electrical circuit can be described using trigonometric functions. Finding the general solution allows engineers and physicists to predict the behavior of these systems over time.
